A Non-local Quasi-Equilibrium State in the Bhatnagar–Gross–Krook Boltzmann Equation for Thermo-hydrodynamics: Conservation Laws, The Boltzmann H-Theorem, and the Fluctuation-Dissipation Theorem

The Bhatnagar–Gross–Krook (BGK) Boltzmann equation with the Maxwellian–Boltzmann-type equilibrium state leads to the set of thermo-hydrodynamic equations such as the continuity, the Navier–Stokes, and the heat-transfer equations in the scaling limit. In this study, using an analytical technique of the kinetic generator, we employ a non-local formulation for the equilibrium state, called a quasi-equilibrium state, to derive the thermo-hydrodynamic equations with flexible choices of transport coefficients and the equation of state (EOS). It is formulated with the non-local macroscopic quantities so that the long-range interaction is explicitly included. According to the new formulation, the consistency with the conservation laws, the Boltzmann H-theorem, and the fluctuation-dissipation theorem is investigated.